4CC30 – Design Principles
Guided Self-study 1
Exercise 1:
Guided Self-study 1
Q2 2021-2022
Design for stiffness and low weight 1
Bending stiffness of a diving board
Figure 1.1 shows a person that would like to use a diving board to take a splash in the pool.
Figure 1.1: A cool person on a diving board
The bending stiffness at the end of the diving board is 1 × 104 N/m. Calculate how much deflection will occur
when a person weighing 80 kg (800 N) is:
1.a)
Standing on the end of the diving board
1.b)
Standing in the center of the diving board
Exercise 2:
Stiffness under different loads
To get some more grip on the effects of different loads on an object’s
stiffness, we will have a look at a simply clamped plate such as in
figure 1.2.
The mechanical properties of the plate can be found in table 1.1.
Table 1.1: Properties of the plate
Quantity
Symbol Amount
Length
l
80
Height
h
40
Thickness
t
1
Young’s modulus
E
210
Shear modulus
G
80
Unit
mm
mm
mm
GPa
GPa
l
h
z
y
t
x
Figure 1.2: A simply clamped plate
Based on these properties, the various moments of area can be
determined using equations 1 and 2. Variables a and b need to be
switched to the relevant dimensions l, h, and t, depending on the
situation.
a · b3 mm4
12
a · b3 Ip =
mm4
3
I =
(1)
(2)
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4CC30 – Design Principles
2.a)
Guided Self-study 1
Q2 2021-2022
Axial load
The first loadcase under consideration is a tensile force in the axial
direction of the plate, as per figure 1.3. What stiffness does the
plate have under a load like this?
Bonus: What is the risk when pointing the force in the other direction? Does this impact the stiffness?
F
z
y
x
Figure 1.3: A plate loaded in tension
2.b)
Bending load due to a force
The force now acts upon the top end of the plate as per
figure 1.4. What is the resulting stiffness for this force?
Bonus: What would the stiffness of the plate be when
the force acted in x-direction?
F
z
y
x
Figure 1.4: A plate loaded in bending
due to a force
2.c)
Bending load due to a moment
We will now attempt to bend the plate in-plane by applying a
moment around the x-asis as per figure 1.5. What is the stiffness
of the plate against rotations?
Bonus: What would the stiffness of the plate be when applying the
moment around the z-axis?
M
z
y
x
Figure 1.5: A plate loaded in (inplane) bending due to a moment
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4CC30 – Design Principles
2.d)
Guided Self-study 1
Q2 2021-2022
Torsional load
We will now apply a moment to the plate in a different direction,
as seen in figure 1.6, which will load the plate in torsion. What is
the torsional stiffness of the plate?
T
z
y
x
Figure 1.6: A plate loaded in torsion
2.e)
Stiffness
Going by the answers you’ve found in the previous questions, in what directions is the plate stiff and in what
directions is it compliant?
Exercise 3:
Stiffnesses in series or parallel
Below you can find a number of scenarios with different stiffnesses working on a body. Indicate, per scenario,
if the stiffnesses are connected in series or parallel. Explain your reasoning.
3.b)
3.a)
3.c)
c1
c1
c1
c2
F
F
c2
c2
3.d)
c1
F
3.f)
3.e)
c2
F
c1
F
c2
T
F0
c2
c1
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4CC30 – Design Principles
Exercise 4:
Guided Self-study 1
Q2 2021-2022
Transmission ratios
Figure 1.7 displays a schematic view of a French fry-cutter.
F
c
l1
l2
Figure 1.7: An abstract fry-cutter
This drawing contains a hinge, a potato (spring) with stiffness c = 100 N/mm, an arm with total length L,
divided in l1 and l2 , which are 100 mm and 300 mm respectively.
4.a)
The way there
When you push the arm down at its end by u = 40 mm, how far will you compress the spring?
4.b)
Reactions
How big will the reaction force of the potato be?
4.c)
The way back
How much force should you apply at the end of the arm to hold this reaction force?
4.d)
Transmission ratios and stiffness
What is the transmission ratio i between the end of the arm and the spring, and what effect does this have on
the spring stiffness you feel at the end of the arm?
Exercise 5:
Stiffness and preloading
To make a bolted connection such as the one in figure 1.8 resistant against
fatigue, it is often desirable to apply a high preload. The bolt has a
stiffness of cbolt , and the clamped part (in orange) has a stiffness ctube .
For simplicity we will assume that only the free bolt length l and the
clamped tube have finite stiffness. Because the bolt is relatively long
and is screwed thight, it will behave like a spring. The tube the bolt is
threaded through will also (slightly) compress.
F/2
F/2
l
Figure 1.8: A bolted connection
5.a)
Series or parallel
Explain if this way of preloading can be considered as a series or parallel connection of the two stiffnesses.
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4CC30 – Design Principles
5.b)
Guided Self-study 1
Q2 2021-2022
Stiffness without preload
What stiffness does the force on the top block feel when no preload has been applied to the bolt?
5.c)
Stiffness with preloading
What stiffness does the force on the top block feel when a preload has been applied, assuming the load is smaller
than the preload?
5.d)
Overcoming the preload
Draw an F ,s-diagram for the force F between 0 N and 6 kN. You may assume that cbolt = 1000 N/mm and
ctube = 4000 N/mm. The preload force F0 equals 5 kN.
Exercise 6:
Optimizing material usage
As exercise 2 made clear, the direction of the load is very important to the stiffness of the design. For the next
figures, try to sketch some ideas on how to better use the available material to maximize stiffness for the applied
load.
Material needs to be connected to the fixed world somewhere, and also the point where the force is applied
needs to contain material. You do not need to take into account other effects such as buckling, tolerances, or
loads in different directions than indicated.
6.b)
6.a)
F
F
F
6.c)
The front fork of a bicycle is typically manufactured as a hollow tube, whereas it would be a lot easier to make
it a solid bar. Explain why, at least for bicycles, front forks are still typically manufactured hollow.
Page 5
4CC30 – Design Principles
Guided Self-study 1
Q2 2021-2022
Hints
Exercise 1:
Bending stiffness of a diving board
Hint 1: What does a diving board look like schematically? This could help setting up the exercise.
Hint 2: Using only the (given) stiffness and mass should be enough in exercise 1.a).
Hint 3: What is the effect of changing length on the stiffness?
Exercise 2:
Stiffness under different loads
Hint 1: The necessary equations should be known from early mechanics courses, and can be found again in the
book ”Design Principles for Precision Mechanisms” by Herman Soemers, in paragraph 1.4.2.
Hint 2: All the information you need to calculate the stiffnesses has been given, except the equations you will
need to add these to.
Exercise 3:
Stiffnesses in series or parallel
Hint 1: How does each stiffness behave individually under the applied load?
Hint 2: What would happen if the stiffnesses are significantly different from each other?
Exercise 4:
Transmission ratios
Hint 1: To see a fry-cutter like this in action you can try getting some fries at Friture Zwerts, situated on the
Boschdijk in Eindhoven.
Hint 2: A force on a certain arm leads to a moment, and moments should be balanced.
Exercise 5:
Stiffness and preloading
Hint 1: The exact amount of preload applied does not affect the stiffness.
Hint 2: What is the effect of overcoming the preload forces in an F ,s-diagram?
Exercise 6:
Optimizing material usage
Hint 1: What is the most optimal way to transfer a force?
Hint 2: In which directions is a plate stiff?
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